a) Describe the form of the coordinates of all points that are equidistant from the x- and y-axes.
b) Describe the form of the coordinated of ll points that are equidistant from the x-, y- and z-axes.
a) The points that are equidistant from the x- and y-axes have coordinates of the form (a, a) or (-a, a), where a is a real number. This means that the x-coordinate and y-coordinate of these points are equal in magnitude, but can have opposite signs.
b) The points that are equidistant from the x-, y-, and z-axes have coordinates of the form (a, a, a) or (-a, a, a), where a is a real number. This means that the x-coordinate, y-coordinate, and z-coordinate of these points are all equal in magnitude, but can have opposite signs.
a) To find the form of the coordinates of points that are equidistant from the x- and y-axes, we have to consider the distance formula.
The distance formula between two points in a 2-dimensional plane is given by:
d = √((x2 - x1)² + (y2 - y1)²),
where (x1, y1) and (x2, y2) are the coordinates of the two points, and d is the distance between them.
For points that are equidistant from the x- and y-axes, the distance from any point (x, y) to the x-axis is the same as the distance from (x, y) to the y-axis. Let's assume that distance to be 'r'.
Using the distance formula, the equation becomes:
r = √((x - 0)² + (y - y1)²),
r = √(x² + (y - y2)²).
Simplifying the equation, we get:
x² + (y - y1)² = r²,
x² + (y - y2)² = r².
So, the form of the coordinates of all points equidistant from the x- and y-axes is (x, y) such that their distances from the x-axis and y-axis are equal.
b) To find the form of the coordinates of points equidistant from the x-, y-, and z-axes, we need to consider the distance formula in 3-dimensional space.
The distance formula between two points in a 3-dimensional space is given by:
d = √((x2 - x1)² + (y2 - y1)² + (z2 - z1)²),
where (x1, y1, z1) and (x2, y2, z2) are the coordinates of the two points, and d is the distance between them.
For points that are equidistant from the x-, y-, and z-axes, the distance from any point (x, y, z) to each axis is the same. Let's assume that distance to be 'r'.
Using the distance formula, the equation becomes:
r = √((x - x1)² + (y - y1)² + (z - z1)²),
r = √((x - x2)² + (y - y2)² + (z - z2)²).
Simplifying the equation, we get:
(x - x1)² + (y - y1)² + (z - z1)² = r²,
(x - x2)² + (y - y2)² + (z - z2)² = r².
So, the form of the coordinates of points equidistant from the x-, y-, and z-axes is (x, y, z) such that their distances from the x-axis, y-axis, and z-axis are equal.